Q82 / 180 · COMEDK 2018 · Mathematics

The value of $(0.2) \log _{\sqrt{5}}\left(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty\right)$ is

  • Correct answer
    A
    4
  • B
    $\frac{1}{4}$
  • C
    2
  • D
    $\frac{1}{2}$

Solution

$(0.2)$ $=(0.2) \quad \log _{\sqrt{5}} \frac{1 / 4}{1-\frac{1}{2}}$ $=(0.2)^{\log _{\sqrt{5}}\left(\frac{1}{2}\right)}$ $=\left(\frac{1}{5}\right)^{\log _{\sqrt{5}}\left(\frac{1}{2}\right)}=\left(\frac{1}{2}\right)^{2 \log _{5} \frac{1}{5}}$ $=\frac{1^{-2 \log _{5} 5}}{2}=\left(\frac{1}{2}\right)^{-2}$ $=\left(2^{-1}\right)^{-2}=2^{2}=4$

Hence, the correct answer is option: 'A'.

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